# NCERT Exemplar Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Exemplar Class 12 Maths Chapter 11 Three Dimensional Geometry are part of NCERT Exemplar Class 12 Maths. Here we have given Exemplar Problems for Class 12 Maths Chapter 11 Three Dimensional Geometry PDF.

## NCERT Exemplar Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Exemplar Class 12 Maths Chapter 11 Three Dimensional Geometry Solutions is given below.

5. Prove that the line through A(0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9,4) and D(- 4,4,4).
Sol. We know that, the equation of a line that passes through two points (x1, y1, z1) and (x2, y2, z2) is

7. Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3,4) and 5(4, 5, 8) at right angles.

8. Find the equation of a plane which is at a distance 3 √3 units from origin and the normal to which is equally inclined to coordinate axis.

9. If the line drawn from the point (-2, -1,-3) meets a plane at right angle at the point (1, -3, 3), find the equation of the plane.
Sol. Since, the line drawn from the point B(-2, -1,-3) meets a plane a plane at right angle at the point 4(1,-3,3).
So, the plane passes through the point 4(1, -3, 3)

10. Find the equation of the plane through the points (2, 1, 0), (3, -2, -2) and (3, 1, 7).
Sol. We know that, the equation of a plane passing through three non-collinear points (x1. y1, z1), (x2, y2, z2) and (x3, y3, z3) is

14. O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.

18. Find the length and the 1oo of perpendicular from the point (1,3/2,2) to the plane 2x – 2y + 4z +5 = 0.
Solution:
Given plane is 2x – 2y + 4z + 5 = O and point (1,3/2,2)
The direction ratios of the normal to the plane aie 2, -2, 4
So, the equation of the line passing through (1,3/2,2) and direction ratios are equal to the direction
ratios of the normal to the plane i.e. 2. -2, 4 is

19. Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y-z = 0.

20. Find the equation of the plane through the points (2,1, -1) and (-1,3,4), and perpendicular to the plane x-2y + 4z = 10.

22. Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y- z + 5 = 0.

27. Show that the straight lines whose direction cosines are given by 2l+2m -n=0 and mn + nl + lm = 0 are at right angles.

Objective Type Questions

30. If the directions cosines of a line are k, k, ‘k, then

33. The reflection of the point (α, β,γ) in the xy-plane is
(a) (α, β,0) (b) (0,0, γ) (c) (-α, -β, γ) (d) (α, β,-γ)
Sol. (d) In xy-plane, the reflection of the point (α, β,γ) is (α, β,-γ)
34. The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, -1), C(4, 5, 0) and D(2, 6,2) is equal to
(a) 9 sq. units (b) 18 sq. units (c) 27 sq. units (d) 81 sq. units

35. The locus represented by xy + yz = 0 is
(a) A pair of perpendicular lines (b) A pair of parallel lines
(c) A pair of parallel planes (d) A pair of perpendicular planes

Fill in the Blanks Type Questions

True/False Type Questions

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